All of the 5th grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$3.00$ each for students, and the group paid $$32.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$34.00$ each for teachers and $$9.50$ each for students, and the group paid $$115.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+3y = 32}$ ${34x+9.5y = 115.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-12y = -128}$ ${34x+9.5y = 115.5}$ Add the top and bottom equations together. $ -2.5y = -12.5 $ $ y = \dfrac{-12.5}{-2.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {8.5x+3y = 32}$ to find $x$ ${8.5x + 3}{(5)}{= 32}$ $8.5x+15 = 32$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 5}$ into $ {34x+9.5y = 115.5}$ and get the same answer for $x$ ${34x + 9.5}{(5)}{= 115.5}$ ${x = 2}$ There were $2$ teachers and $5$ students on the field trips.